We classify all possible allowed constitutive relations of relativistic
fluids in a statistical mechanical limit using the Schwinger-Keldysh effective
action for hydrodynamics. We find that microscopic unitarity enforces genuinely
new constraints on the allowed transport coefficients that are invisible in the
classical hydrodynamic description; they are not implied by the second law or
the Onsager relations. We term these conditions Schwinger-Keldysh positivity
and provide explicit examples of the various allowed terms.

In non-relativistic quantum theories the Lieb-Robinson bound defines an
effective light cone with exponentially small tails outside of it. In this work
we use it to derive a bound for the time evolution of the correlation function
of two local disjoint observables if the initial state has a power-law decay.
We show that the exponent of the power-law of the bound is identical to the
initial (equilibrium) decay. We explicitly verify this result by studying the
full dynamics of the susceptibilities and correlations in the exactly solvable
Luttinger model after a sudden quench from the non-interacting to the
interacting model.

Let $\mathfrak{g}$ be a simply laced Lie algebra, $\widehat{\mathfrak{g}}_1$
the corresponding affine Lie algebra at level one, and
$\mathcal{W}(\mathfrak{g})$ the corresponding Casimir W-algebra. We consider
$\mathcal{W}(\mathfrak{g})$-symmetric conformal field theory on the Riemann
sphere. To a number of $\mathcal{W}(\mathfrak{g})$-primary fields, we associate
a Fuchsian differential system. We compute correlation functions of
$\widehat{\mathfrak{g}}_1$-currents in terms of solutions of that system, and
construct the bundle where these objects live. We argue that cycles on that
bundle correspond to parameters of the conformal blocks of the W-algebra,
equivalently to moduli of the Fuchsian system.

We study complex CFTs describing fixed points of the two-dimensional
$Q$-state Potts model with $Q>4$. Their existence is closely related to the
weak first-order phase transition and walking RG behavior present in the real
Potts model at $Q>4$. The Potts model, apart from its own significance, serves
as an ideal playground for testing this very general relation. Cluster
formulation provides nonperturbative definition for a continuous range of
parameter $Q$, while Coulomb gas description and connection to minimal models
provide some conformal data of the complex CFTs. We use one and two-loop
conformal perturbation theory around complex CFTs to compute various properties
of the real walking RG flow. These properties, such as drifting scaling
dimensions, appear to be common features of the QFTs with walking RG flows, and
can serve as a smoking gun for detecting walking in Monte Carlo simulations.
The complex CFTs discussed in this work are perfectly well defined, and can
in principle be seen in Monte Carlo simulations with complexified coupling
constants. In particular, we predict a pair of $S_5$-symmetric complex CFTs
with central charges $c\approx 1.138 \pm 0.021 i$ describing the fixed points
of a 5-state dilute Potts model with complexified temperature and vacancy
fugacity.

We investigate the validity of Luttinger's theorem (or Luttinger sum rule) in
two scale-invariant fermionic models. We find that, in general, Luttinger's
theorem does not hold in a system of fermions with power-law Green functions
which do not necessarily preserve particle-hole symmetry. However, Ref.
\cite{Blagoev1997,Yamanaka1997} showed that Luttinger liquids, another
scale-invariant fermionic model, respect Luttinger's theorem. To understand the
difference, we examine the spinless Luttinger liquid model. We find two
properties which make the Luttinger sum rule valid in this model: particle-hole
symmetry and $\mathrm{Im} G(\omega=0,-\infty)=0$. We conjecture that these two
properties represent sufficient, but not necessary, conditions for the validity
of the Luttinger sum rule in condensed matter systems.

It has been recently shown that the deformation of an arbitrary
two-dimensional conformal field theory by the composite irrelevant operator $T
\bar T$, built from the components of the stress tensor, is solvable; in
particular, the finite-size spectrum of the deformed theory can be obtained
from that of the original CFT through a universal formula. We study a similarly
universal, Lorentz-breaking deformation of two-dimensional CFTs that posess a
conserved $U(1)$ current, $J$. The deformation takes the schematic form $J \bar
T$ and is interesting because it preserves an $SL(2,\mathbb{R}) \times U(1)$
subgroup of the original global conformal symmetries. For the case of a purely
(anti)chiral current, we find the finite-size spectrum of the deformed theory
and study its thermodynamic properties. We test our predictions in a simple
example involving deformed free fermions.

We explore in detail the implementation of arbitrary abelian and non-abelian
symmetries in the setting of infinite projected entangled pair states on the
two-dimensional square lattice. We observe a large computational speed-up;
easily allowing bond dimensions $D = 10$ in the square lattice Heisenberg model
at computational effort comparable to calculations at $D = 6$ without
symmetries. We also find that implementing an unbroken symmetry does not
negatively affect the representative power of the state and leads to identical
or improved ground-state energies. Finally, we point out how to use symmetry
implementations to detect spontaneous symmetry breaking.

We consider time-of-flight measurements in split one-dimensional Bose gases.
It is well known that the low-energy sector of such systems can be described in
terms of two compact phase fields $\hat{\phi}_{a,s}(x)$. Building on existing
results in the literature we discuss how a single projective measurement of the
particle density after trap release is in a certain limit related to the
eigenvalues of the vertex operator $e^{i\hat{\phi}_a(x)}$. We emphasize the
theoretical assumptions underlying the analysis of "single-shot" interference
patterns and show that such measurements give direct access to multi-point
correlation functions of $e^{i\hat{\phi}_a(x)}$ in a substantial parameter
regime. For experimentally relevant situations, we derive an expression for the
measured particle density after trap release in terms of convolutions of the
eigenvalues of vertex operators involving both sectors of the two-component
Luttinger liquid that describes the low-energy regime of the split condensate.
This opens the door to accessing properties of the symmetric sector via an
appropriate analysis of existing experimental data.

We provide a pedagogical review on the calculation of highly excited
eigenstates of disordered interacting quantum systems which can undergo a
many-body localization (MBL) transition, using shift-invert exact
diagonalization. We also provide an example code at
https://bitbucket.org/dluitz/sinvert_mbl/. Through a detailed analysis of the
simulational parameters of the random field Heisenberg spin chain, we provide a
practical guide on how to perform efficient computations. We present data for
mid-spectrum eigenstates of spin chains of sizes up to $L=26$. This work is
also geared towards readers with interest in efficiency of parallel sparse
linear algebra techniques that will find a challenging application in the MBL
problem.

One of the main challenges in simulations on Lefschetz thimbles is the
computation of the relative weights of contributing thimbles. In this paper we
propose a solution to that problem by means of computing those weights using a
reweighting procedure. Besides we present recipes for finding parametrizations
of thimbles and anti-thimbles for a given theory. Moreover, we study some
approaches to combine the Lefschetz thimble method with the Complex Langevin
evolution. Our numerical investigations are carried out by using toy models
among which we consider a one-site z^4 model as well as a U(1) one-link model.